A 3-D Finite Element Model for Gas-AssistedInjection Molding: Simulations and

Experiments

G.A.A.V. Haagh, G.W.M. Peters†, F.N. vandeVosse,andH.E.H.Meijer

EindhovenUniversityof Technology, Departmentof MechanicalEngineeringP.O. Box 513,5600MB Eindhoven,TheNetherlands

(Presentedat the14thAnnualMeetingof thePolymerProcessingSociety, Yokohama,Japan,June8-12,1998)

Presentaffiliation: UnileverResearchVlaardingen,TheNetherlands.

†Author to whomcorrespondenceshouldbeaddressed.

A 3-D Finite Element Model for Gas-Assisted Injection Molding: Simulations andExperiments

G.A.A.V. Haagh, G.W.M. Peters†, F.N. vandeVosse,andH.E.H.Meijer

EindhovenUniversityof Technology, Departmentof MechanicalEngineeringP.O. Box 513,5600MB Eindhoven,TheNetherlands

Abstract

Althoughgas-assistedinjectionmolding(GAIM) hasbeenpracticedin industryfor morethana decade,theprocessis not completelyunderstood,particularlywithrespectto the gaspenetrationmechanism.Consequently, mold designandprocesscontrol are often governedby trial-and-error, and reliable information on the gasdistribution andthefinal productpropertiescanoftenonly beobtainedfrom experi-ments.To gaina betterunderstandingof thegas-assistedinjectionmoldingprocess,we have developeda computationalmodelfor the GAIM process.This modelhasbeensetupto dealwith (non-isothermal)three-dimensionalflow, in ordertocorrectlypredict the gasdistribution in GAIM products. It employs a pseudo-concentrationmethod,in which thegoverningequationsaresolvedon a fixedgrid thatcoverstheentiremold. Both the air downstreamof the polymer front and the gasare repre-sentedby a fictitious fluid thatdoesnot contribute to thepressuredropin themold.Themodelhasbeenvalidatedagainstbothisothermalandnon-isothermalgasinjec-tion experiments.In contrastto othermodelsthathavebeenreportedin theliterature,ourmodelyieldsthegaspenetrationfrom theactualprocessphysics(not from apre-supposedgasdistribution). Consequently, it is able to dealwith the 3-D characterof the process,aswell aswith primary (endof gasfilling) andsecondary(endofpacking)gaspenetration,including temperatureeffectsandgeneralizedNewtonianviscositybehavior.

Presentaffiliation: UnileverResearchVlaardingen,TheNetherlands.

†Author to whomcorrespondenceshouldbeaddressed.

1

Figure1: The principle of gas-assistedinjection molding: polymer injection (top), gasinjection(middle),andpacking(bottom).Light grey = polymermelt,darkgrey= solidifiedpolymer, white= gas/air.

1 Introduction

In Gas-AssistedInjection Molding (GAIM), gasis injectedinto a mold that hasbeenpartially filled with polymer (seeFigure 1). The gasdrives the molten polymer corefurther into themold, until it is filled completely. The penetratinggasleavesa polymerlayerat themold walls, yielding a productwith a polymerskin anda gascore. Thegascaneitherbe injectedthrougha needlein the extrudernozzle,or directly into the moldthroughseparategasinjectionneedles.

After the mold hasbeenentirely filled, gasis usedto transmitthe packingpressureto thepolymerthat is beingcooled.Any shrinkageof thepolymermaterialnearthegaschannelis compensatedfor by anenlargementof thegascore.Onceall polymermaterialhassolidified,thegaspressureis released.Theproductis thenfurthercooleduntil it hasretainedsufficient rigidity to beejectedfrom themold.

Themostimportantcharacteristicof GAIM is thefactthatthepressuredropin thegascoreis negligibly small comparedto thepressuredrop in an equivalentmoltenpolymercore,becausetheviscosityof thegasis roughly108 timessmallerthanthatof thepolymer.Consequently, thepressurecanbeconsideredconstantthroughoutthegascore,andthisaccountsfor most of the advantagesof GAIM, suchas reductionof clamp force, sinkmarksandresidualstresses,andenhancementof designpossibilities.

Theinjectedgastypically penetratesalongthepathof leastflow resistance.This hastwo consequences:first, any thick-walledpartof a mold offersa significantlylower flowresistancethan the thin-walled partsthat usuallymake up a major part of the product.Thegasis, therefore,inclinedto penetrateinsidethethick-walledparts,suchasribs andbosses.GAIM productandmolddesignersshouldaccountfor this fact,eitherby properlyincorporatingribs to leadthegasflow, or by supplyingdesignatedgas-leadingchannels

2

[1, 2]. Althoughthegaswill generallypenetratethroughthethick parts,it mayflow intothinnerpartsassoonasthethick partis completelyfilled. Thesecondconsequenceof thegasfollowing thepathof leastflow resistance,is the inherentinstability of thegasfrontadvancementat — apparentlysymmetric— bifurcationsin themold [3, 4].

Although gas-assistedinjection molding hasbeenpracticedcommerciallyfor morethana decade,the understandingof the characteristicsof the process,particularlywithrespectto the typical flow phenomena,is still laggingbehind. Theseyearsof practicalGAIM experience,mostly gainedfrom trial-and-error, have led to designguidelinesforGAIM moldsandproducts,which, however, do not alwayshave an explicit connectionto the physicsof the process.Consequently, several researcherswho have investigatedtheeffect of severalparameterson theprocessandfinal product,sometimesreportcon-tradictoryconclusions[5]. A morethoroughunderstandingof theprocessis expectedtoestablisha clearconnectionbetweentheseprocessparametersandthe processphysics,andmayhencerevealwhichparametersareimportant(andwhicharenot!).

Themechanismof gaspenetrationis similar to thatof a viscousfluid in a tubebeingdisplacedby anotherfluid of lower viscosity, which hasbeenstudiedfor a long time[6–10]. It hasbeenshown that part of the initially present,moreviscousfluid remainssticking to the tubewall, and that this fraction is determinedby the Capillary number,givenby:

Ca ηUγ (1)

Poslinskietal. [11] wereamongthefirst to connectthis researchto gas-assistedinjectionmolding. In orderto decouplethethermalandtheviscouseffectsongaspenetration,theyperformedboth isothermalgasinjection experimentsandan isothermalanalysisof theproblem. From their dimensionalanalysis,they concludedthat the inertial andgravityeffectscanbe neglected,leaving the Capillary numberasthe sole factorcharacterizingtheflow.

Their experimentson primary gaspenetrationin a Newtonian liquid show that theresidualwall thicknessε, which is definedasthe ratio of the residualskin layer of liq-uid andthetuberadius,approachesa valueof 0.35— correspondingto a cross-sectionalresidualliquid fractionφ 0 58 — for sufficiently largeCapillarynumbers(Ca 102).As Ca is at leastof order103 for thegasinjectionstageof GAIM, they alsoneglectedin-terfacialtensioneffects.Fromfinite elementcomputations,they concludethatε decreaseswith increasingpower-law exponentfor generalizedNewtonianfluids (seeFigure2).

Themaincontributionsof Poslinskiandco-workersaresummarizedin thefollowingconclusions: The residualpolymerwall thicknessin GAIM is determinedby two phenomena:

thepenetrationof agasinto aviscousliquid, andthegrowth of asolid layer. As thesolid layergrowth during theprimarygasinjectionstageis negligibly small,thecontributionof thermalandviscouseffectsto theresidualwall thicknesscanbedecoupled;thefinal residualwall thicknessis thesumof thesolid layer thicknessat the beginning of gasinjection and the liquid polymer layer left behindby thepenetratinggas.

3

Figure2: Residualwall thicknessasa function of the Capillary numberfor Newtonianandpower-law fluids (η η0γn 1). (After Poslinskiet al. [11].)

4

For Newtonianfluids in isothermalflows,theresidualwall thicknessis only a func-tion of the Capillary number. Non-uniformity of the viscosity, either causedbytemperatureor shearrategradients(in the caseof non-Newtonianfluids) furtheraffectstheresidualwall thickness.

Objective

To predictthegasdistributionin aproductandtheresultingproductproperties,numericalsimulationsof GAIM canbe a powerful tool. Moreover, suchsimulationsmay help toincreasetheunderstandingof processfeaturesandtherebycontributeto moldingexperi-ence.We have thereforedevelopeda computationalmodelfor GAIM simulations.It isbasedon thephysicsof theprocess,ratherthanon a sheerdescriptionof thephenomenathat areobserved. Consequently, this modelemploys a three-dimensionalapproach,asgasinjection is governedby three-dimensionalphenomena.Themodelhasprovento atleastqualitatively predictsomephenomenathatarecharacteristicfor thegas-assistedin-jectionmoldingprocess[12–14].Theobjectiveof thispaperis to validatethissimulationprogramexperimentally, by focusingon thepredictionof thegasdistribution(usuallyex-pressedin termsof residualwall thickness).Thenumericalresultswill becomparedwithexperimentalresultsfor relatively simple,but yet characteristic,gasinjectionconditions.

First, the descriptionof the computationalmodeland its numericalimplementationwill begiven,andtheexperimentalset-upwill bepresented.Subsequently, thesimulationresultswill be comparedwith experimentaldata,from which we will eventuallydrawconclusionswith respectto thevalidity andapplicabilityof themodel.

2 Governing Equations

Ourmodelfor three-dimensionalgas-assistedinjectionmoldingsimulationsis basedonapseudo-concentrationmethodof Thompson[15], which is relatedto theVolumeof Fluid(VOF) method[16]. With this method,the flow problemis solved on a fixed grid thatcoverstheentiremold, sothatelaboratethree-dimensionalremeshingcanbeavoided.Afictitiousfluid is introducedto representboththeair downstreamof thepolymerflow frontandthe injectedgas. The main propertyof this fictitious fluid is that its contribution tothepressurebuild-upin themoldduringfilling is negligible. Therefore,its viscosityis setto a valueat least103 timessmallerthanthe viscosityof thefilling fluid. However, theviscosityof thefictitiousfluid exceedstherealvaluefor air by severalordersof magnitudein orderto keeptheReynoldsnumbersmall,sothatinertiatermsdo not have to betakeninto accountandturbulenceis avoided.Furthermore,thefictitiousfluid is allowedto leavethemold at somespecifiedboundaries.

Theessenceof thepseudo-concentrationmodelis that thedistinctionbetweenfillingfluid andfictitious fluid is madeby labellingfluid particleswith a materiallabelc (beingthepseudo-concentration),which is giventhevaluec 1 for thefilling fluid, andc 0for the fictitious fluid. Nearthe flow front, c is a continuousfunction between1 and0;theflow front itself is determinedby theiso-valueline for c 0 5. This interfaceis kepttrackof by convectingthemateriallabelswith thefluid velocity.

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Table1: Processvariablesexpressedasproductsof dimensionlessvariables(markedwithan asterisk:

) andcharacteristicvalues. (The rate-of-deformationtensorD has

beenscaledwith thelargestcomponentsof ∇ u for ε 1, whichare ∂u∂y and ∂u

∂z.)

x xL y y

H y

εL z z

H z

εL

u uU v v

V v

εU w w

V w

εU

D D U

H ε HL p p

pref

t tτ T T

∆T 0 g g

g0

η ηη0 ρ ρ

ρ0 cp c

pcp0

λ λλ0 α α

α0

Theconservationequationsfor mass,momentum,andinternalenergy read:

∂ρ∂t

∇ ρu 0 (2)

ρ∂u∂t

ρu ∇ u ∇ σ ρg (3)

ρe σ : D ∇ h ρr ρhrRc (4)

Formally, theconservationof momentof momentum,which requiresthat thestressten-sor is symmetric, is addedto this set of equations. Moreover, the use of a pseudo-concentrationmethoddemandsfor an equationfor passive scalarconvection (‘conser-vationof identity’):

c 0 (5)

which statesthateachmaterialparticle(actually:eachinfinitesimalmaterialvolumeele-ment)in themold is labelledwith avaluec thatdoesnotchange.

2.1 Flow problem

In previous publications[12–14], we have shown througha dimensionalanalysisthatthe Navier-Stokesequationsfor both the polymerdomainandthe air domainin oblonggeometries— of which ribs andothergas-leadingchannelsaretypical examples— canbereducedto a stationaryStokesequation:

∇ p ∇2ηD (6)

assumingthecharacteristicvaluesfor injectionmoldingthatarepresentedin Table2. Thevalueη0 104Pas is characteristicfor non-isothermalflow of polymermeltsin injectionmolding,i.e., whenthepolymercoolingis takeninto account.Furthermore,thesolidifiedpolymeris modeledasanextremelyviscousfluid: assoonasthepolymersolidifies,itsviscosity is setto 106Pas, which exceedsthe melt viscosityby, approximately, a factor102–103.

Usually, polymerflow is assumedto be incompressible.However, we will retaintheoriginal continuityequation(2) in orderto beableto computepolymershrinkage.Thus,weconsiderthepolymerdensityto beagivenfunctionof pressureandtemperature:

ρ ρp T (7)

6

Table2: Characteristicvaluesof the processvariablesfor thermoplasticinjection mold-ing.

variable unit characteristicvaluepolymer air

ρ0 kgm 3 103 1η0 Pas 104 10 5

cp0 Jkg 1K 1 103 103

λ0 Wm 1K 1 10 1 10 2

α0 K 1 10 5 10 3

L m 10 1

H m 10 2

U ms 1 10 1

g0 ms 2 10γ0 Nm 1 10 2∆T 0 K 102

Neglectingthebulk viscosity[14, 17], thedimensionlessconservationequationfor masscanbewrittenas:

Sr∂ρ∂t

∇ ρu 0 (8)

Boundary conditions

WedefineadomainΩ coveringthemold,with boundariesΓe, Γw, andΓv designatingthemoldentrance,themoldwalls,andtheair vents.At themoldentrance,eithertheinjectionflow rateor thenormalstress(i.e., theinjectionpressure)is prescribed.Whereverthemoldwall is coveredwith polymer, ano-slipconditionis imposed.

As ano-slipboundaryconditionin theair wouldpreventthepolymerfrom contactingthemold wall, we have chosento prescribea freeslip conditiondownstreamof theflowfront, thusenablingthecontactpoint to movefreely [13]. Hence,theboundaryconditionalong the mold walls is a function of the type of material,which is indicatedby thematerial label c. This hasbeenimplementedby using an adjustableRobin boundaryconditionfor the (dimensionless)velocity andstresscomponentsut andσt in tangentialdirection:

aut σt 0 x

Γw Γv (9)

in which thedimensionless‘Robin penaltyparameter’a is definedas

a ac

large

104 if c

0 5: no slip0 if c 0 5: freeslip

(10)

7

Themold walls areimpermeable,exceptat theair ventsΓv wheretheair is allowedto leave themold, yielding thefollowing boundaryconditionsfor thevelocity andstresscomponentun andσn in normaldirection:

un 0 x Γw (11a)

aun σn 0 x Γv (11b)

in which a is againgiven by equation(10) (provided that ‘slip’ shouldbe replacedby‘leakage’).

Interfacial conditions

From a physicalpoint of view, two moreboundaryconditionshold for the flow front:immiscibility and conservation of momentum. The immiscibility condition is alreadyimpliedby the‘conservationof identity’ (equation5). Theconservationof momentumattheinterfaceis expressedin dimensionlessform as[13]:

σ2 σ1 n12 0 (12)

in which thesubscripts1 and2 denotethepolymerandthefictitious fluid. Thisconditionalreadytakencareof by theoverall conservationof momentumequation,sincethemate-rial propertiesarecontinuousfunctionsof c at the interface.As a result,thephenomenaat theflow front havealreadybeentakeninto accountin theproperway.

2.2 Temperature problem

Assumingthat neither the thermal radiation r nor the reactionheathr plays a role ininjectionmoldingof thermoplastics,andapplyingFourier’s law h λ∇ T, equation(4)canbereducedto:

ρe σ : D ∇ h (13)

Substitutiontheappropriateconstitutiveequationsin equation(13)yields:

ρcpT 2ηD : D ∇ λ∇ T

αT p (14)

in which theheatcapacitycoefficient cp andtheheatconductioncoefficient λ aregivenby:

cp cpp T (15)

λ λp T (16)

Introducingthedimensionlessvariablesgivenin Table1 into this equationyields thedimensionlesstemperatureequation:

1Fo

ρcp∂T∂t

εPeρcpu ∇ T 2BrηD : D

∇ λ∇ T εBrSr

GcαT

∂p0

∂t εBr

GcαTu ∇ p0 (17)

8

from which theasterisk() indicatingdimensionlessvariableshasbeenremovedfor con-

venience.Thedimensionlessnumbersaredefinedas:

Fo λ0τρ0cp0H2 Fouriernumber (18a)

Pe ρ0cp0UH

λ0Pecletnumber (18b)

Br η0U2

λ0∆T 0 Brinkmannumber (18c)

Sr LτU

Strouhalnumber (18d)

Gc 1α0

∆T 0 Gay-Lussacnumber (18e)

Thecharacteristictime τ in thetemperatureequationis typically thetime in which eitherthe temperatureor the pressurechangesin the orderof its magnitude.For the injectionstage,cooling of (hot) polymermelt contactinga (cold) mold wall is roughly estimatedto take about1 second,so that τ 1s for this stage.Thepackingstageis characterizedby a fast increaseof the pressure,for which τ is of the orderof 0 1s. The temperaturedecreasesratherslowly in thecoolingstage,soτ is estimatedto be10s.

As for theflow problem,a distinctionbetweenthepolymerandtheair domainhastobemade,in orderto determinewhich termsof thetemperatureequationareimportantineachdomain.

Polymer domain

Determiningthedimensionlessnumbersdefinedin equation(18) from thecharacteristicvaluesfor thepolymerasgivenin Table2, andsubstitutingtheseinto thedimensionlesstemperatureequation(17)yieldstheordersof magnitudeof thedifferentterms:

1Fo

ρcp∂T∂t

103τ ! 1

εPeρcpu ∇ T "small

" 2BrηD : D

10

∇ λ∇ T 1

εBrSrGc

αT∂p0

∂t 10! 3τ ! 1

εBrGc

αTu ∇ p0 10! 3

(19)

The orderof magnitudeof the convictive term (the secondterm on the left handside)is generallysmallbecauseof thevelocity u andthe temperaturegradient∇ T aremostlyperpendicular. Moreover, the largest temperaturegradientis found nearthe wall in atemperatureboundarylayer of a limited thicknessδ H, in which the velocity will beverysmall.This termis only expectedto besignificantnearsharpgeometricaltransitions,suchascornersandflow contractions.

With the givenestimatesfor the characteristictime for thedifferentmolding stages,the last two termsof equation(19) canbe neglectedfor all injection stages,leaving thefollowing temperatureproblemto besolvedfor thepolymerdomain:

1Fo

ρcp∂T∂t

εPeρcpu ∇ T 2BrηD : D ∇ λ∇ T (20)

9

which is anordinaryconvection-diffusionequation.As a matterof fact,thesecond(con-vection)andthird (dissipation)termcanbeneglectedaswell in thecoolingstage,sincethevelocity is approximatelyzerofor thatstage.

Air domain

A similarexerciseyieldstheorderof magnitudeof thetemperatureequationtermsfor theair domain:

1Fo

ρcp∂T∂t

10τ ! 1

εPeρcpu ∇ T "small

" 2BrηD : D

10! 6

∇ λ∇ T 1

εBrSrGc

αT∂p0

∂t 10! 6τ ! 1

εBrGc

αTu ∇ p0 10! 6

(21)

Discardingtheirrelevanttermsresultsin:

1Fo

ρcp∂T∂t

εPeρcpu ∇ T ∇ λ∇ T (22)

whichis identicalto theresultingequationfor thepolymerdomain,exceptfor themissingviscousdissipationterm. However, this term needsa slight re-examination,sinceweartificially increasedthe air viscositywhenwe introducedthe fictitious fluid. Thus,forthe fictitious fluid, the Brinkmannumberis of order10 2, indicating that the useof afictitiousfluid will hardlyaffect thetemperaturesolution.

Consequently, applyingequation(20)ontheentirecomputationaldomainwouldyieldasufficiently accuratetemperaturesolution.However, theviscousdissipationtermshouldbesetto zerofor theair domain,sincethecomputedvelocity field in thefictitious fluiddoesnot representtheactualair velocitydistribution.

Initial and boundary conditions

As an initial conditionfor the temperatureproblem,a temperaturefield over the entiredomainis imposed:

T T0x x Ω t 0 (23)

At theinjectiongateΓe, theinjectiontemperatureis prescribed:

T Tet x Γe t 0 (24)

Theboundaryconditionsat themold walls andair ventscaneitherbea constanttemper-ature(Dirichlet boundarycondition):

T Twt x

Γw Γv # t 0 (25)

10

or aprescribedheatflux (Biot or Robinboundarycondition):

λ∂T∂n

hwT Tw x

Γw Γv # t 0 (26)

in which n is thenormalvectoron themold wall, hw is theeffective heattransfercoeffi-cientfrom thepolymerto thecoolingmedium,andTw is thewall temperature.

Sincetheactualtemperaturesatthemoldwall arerarelyknown accurately, acompletethermalanalysisof the mold (including the cooling channelsetc.) would yield a morereliablemolding simulation. However, suchan extensionof the analysisis beyond thescopeof this paper.

At theair/polymerinterface(s),thetemperatureis continuous.Solvingthetemperatureequation(20)onafixedgrid will inherentlysatisfythis condition.

2.3 Material label convection problem

Themateriallabelsthatareusedto distinguishpolymerfrom air, areconvectedthroughthe mold with velocity u, while maintainingtheir ‘identity’ accordingto equation(5).Hence,in a Euleriancoordinatesystem,a pure(passive scalar)convectionequationde-scribestheevolutionof themateriallabeldistribution; in its dimensionlessform, it reads:

Sr∂c∂t

u ∇ c 0 (27)

Initially, c is setto zeroin theentiredomainΩ, andonly boundaryconditionsat theflowentranceareneeded:

c 0 x Ω t 0 (28a)

c 1 x Γe 0 t tgas (28b)

c 0 x Γe t tgas (28c)

in which tgas is thetimewhengasis injected.Theboundaryconditionfor theconvectionequationcanalsobethetime of entrance

(injection time) or oneof the entrancecoordinates.Equation(27) can thenbe usedtoperformparticletrackingto visualizetheflow [18].

Thematerialproperties,asthey appearin theStokesequation,cannow bedeterminedlocally asafunctionof themateriallabel.Themoldfilling problemcanthusbesimulatedby solvingequations(6) and(27),andupdatingthematerialpropertiesatevery timestep.

2.4 Numerical methods

The continuity equation(8) and the Stokes equation(6) are solved using a standardGalerkin finite elementmethod. Unlike in the purely incompressiblecase,the systemof elementequationsfor thecontinuityequationhasanon-zeroright handsidevector, asit takesinto accountthelocal time derivativeof thedensitythrougha first orderapproxi-mationin time.

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Thematerialparametersaredefinedasdiscontinuousfunctionsof themateriallabels,e.g., for viscosity:

η ηc

ηpolymer if c

0 5ηfictitious if c 0 5 (29)

The temperatureand the label convection equationsare solved with the finite ele-mentmethodusingan StreamlineUpwind Petrov-Galerkinscheme.A finite differenceθ-methodwasusedfor thetemporaldiscretisation.Themateriallabelsareroundedoff toeitherunity or zeroeverywhere,exceptin theelementscontainingtheflow front, wheretheoriginal valuesof themateriallabelsareretained.Hence,oscillationsin themateriallabelfield aresuppressed,andtheflow front is trackedmoreaccurately.

The model,asit hasbeendescribedin theprevious sections,hasbeenimplementedin thefinite elementpackageSEPRAN[19]. A morecomprehensive explanationof thenumericalmethodscanbefoundin references[12–14].

3 Experimental methods and materials

To validatethecomputationalmodelfor gas-assistedinjectionmolding,a numberof gasinjectionexperimentsweresetup. As arguedin the introduction,the residualthicknessconsistsof a solidified layer (governedby thermaleffects) anda melt layer (governedby viscousbehavior). Theviscouscontribution is investigatedseparatelyby performingtwo typesof experiments:isothermalexperiments,in which the temperatureeffectsaredisregarded;andnon-isothermalexperiments,in which thethermalconditionsareusedtoaffect theviscositybehavior.

Consequently, we built two experimentalset-upsfor our validationexperiments:onefor gasinjectionexperimentsin aaxisymmetriccylinder (in thetraditionof Poslinskiandco-workers[11]), andanotheronefor gasinjectionin aplaquewith a rib.

3.1 Cylinder set-up

The experimentalset-upfor gas injection experimentsconsistsof two hollow coaxialcylindersseparatedby a spiralgroove (Figure3). Initially, thecavity is filled with 4mmthick tabletsof polystyreneto aheightthatis sufficient to preventgasbreakthrough(usu-ally 70–80%of the cylinder height). Then the cylinder is heatedby pumpinghot oilthroughthespiralgroove. Coolingthecylinder is doneby switchingto coldoil. By stack-ing alternatinglyblackandyellow polystyrenetabletsinto thecylinder, theflow patternscanbevisualized.

Five type J (iron/constantan)thermocouplesareusedto monitor the temperatureatapproximately3mmfrom theinnermoldwall. In eachexperiment,thecylinder is heatedlong enoughfor the polystyreneto obtaina constanttemperature.Nitrogengascanbeinjectedthrougha pressurevalveeitherbeforecooling(isothermalgasinjection)or afterallowing for a stagnantpolymerlayerto develop(non-isothermalgasinjection). Thegaspressureis in theorderof afew baronly, andgasinjectionmaytakefrom afew secondstoa few minutesto fill theentirecylinder. Onemight objectthatthepressuresusedin theseexperimentsaremuchlower thanthegasinjectionpressuresthatarecommonin GAIM

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insulation

oil 0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1

Figure3: Experimentalset-upfor GAIM experimentsin anaxisymmetriccylinder.

13

practice.However, it wasadvocatedin theintroductionthattheresidualwall thicknessisonly afunctionof theviscositybehavior andtheCapillarynumber, andnotof thepressure.

After the mold hascooledsufficiently, thesampleis pressedout of the cylinder andcut in half. In this way, thegasdistribution andmelt deformationcanbevisualizedandtheresidualwall thicknesscanbemeasured.

3.2 Plaque-with-rib set-up

Theplaque-with-ribset-up,which is depictedin Figure4(a),representsacommonlyusedGAIM geometry, asribs oftenserve asgas-leadingchannels.Severalrib geometriesandtheireffecton thegascoresizehavebeenreviewedin theliterature[2, 20], andtheeffecton productstrengthandstiffnesshasbeeninvestigated[21]. The rib geometryin ourset-uphasbeendesignedin accordancewith theguidelinesgivenby Rennefeld[2].

The principle of the plaque-with-ribset-upis similar to the cylinder set-up: pre-moldedpolystyreneinserts(in threeparts: plaque,rib foot, andrib) areput inside themold. Themoldisclosedandsubmergedintoahotoil bath,in ordertomeltthepolystyreneinserts.After themoldhasattainedahomogeneoustemperature(whichis measuredastheoil temperature),gascanbeinjected.Oncetheprocessis completed,themold is cooledin a cold oil bath(i.e., at room temperature),after which the specimencanbe removedfrom the mold. For non-isothermalexperiments,the gasis injectedafter the mold hasbeensubmergedinto the cold bathfor a specifiedtime. The specimenis cut into slicesperpendicularto its lengthdirection,andtheresidualwall thicknessesaremeasured.

Although our set-upenablesisothermalgasinjection experiments,it doeslack oneparticularfeatureof a real injectionmoldingmachine:thepolymermelt insidethemoldcannotbepressurizedbeforetheactualgasinjection. It appearedthat,asa resultof this,the gasdid not always penetrateinto the molten polystyrene,but often found its waytowardsthe mold walls, alongwhich it escapedto the air vents. In particular, whenthemoldwassubmergedinto thecoldoil bath(for non-isothermalexperiments),thepolymershrinkagedefinitelyprovidedsucha‘shortcircuit’ from theinjectiongateto theair vents.As a consequence,non-isothermalgasinjectionexperimentsturnedout to beimpossiblewith thisset-up,andthegaspenetrationwashardto controlin theisothermalexperiments.

3.3 Material properties

Thepolystyreneusedin theexperimentsis Styron678Efrom DOW Chemical.Theshearviscosityof this polystyreneis describedby a7-constantCrossmodel[22]:

ηp T D 2 η0

1

η0 3 II 2D 354 τ 67 1 n (30a)

in which II 2D is thesecondinvariantof (twice) the rate-of-deformationtensor, andη0 isthezeroshearrateviscositygivenby:

η0T η0

T 6 e ! c1 8 T ! T 9;:

c92 < T ! T 9 (30b)

T 6 T0

s p (30c)

c62 c2

s p (30d)

14

=?>A@CBEDGFH;I

JK=?IL>MFNMDGO?H;>APQFRTS =?UWVTD

@X>MY[Z\PQPH@X>MY

(a)Experimentalset-up

]^_`badc e

fhgii _?`ba5c

jadck

lhm(b) Dimensions. n nEo(0,0,0)

p prq ysz tx

(c) Mesh.

Figure4: Plaque-with-ribgeometry(dimensionsin mm). For reasonsof symmetry, onlyonehalf hasto bemeshed.

15

Table3: Material parametersfor polystyreneStyron 678E from DOW Chemical(after[22]).

(a)Viscosityparameters.

n 0 2520τ 6 Pa 3 080 104

η0T 6 Pas 4 76 1010

T0 K 373c1 25 74c2 K 61 06s K Pa 1 5 1 10 7

(b) Specificvolumeparameters.

melt glass

a0 m3kg 1 9.72 10 4 9.72 10 4

a1 m3kg 1K 1 5.44 10 7 2.24 10 7

B0 Pa 2.53 108 3.53 108

B1 K 1 4.08 10 3 3.00 10 3

Tg0 K 373s K Pa 1 5.1 10 7

(c) Thermalproperties.

melt glass

λ Wm 1K 1 0 17 0 17cp Jkg 1K 1 2289 1785

Table4: Materialparametersfor nitrogengas(atT 273K andp 0MPa; from [26]).

v m3kg 1 0 8λ Wm 1K 1 25cp Jkg 1K 1 1 04 103

The pvT-behavior of thepolystyrenecanbedescribedby a so-calleddouble-domainTait equation[23]:

vp T uv5w

a0s

a1sT Tg0 hyx 1 0 0894ln x 1 p

Bs z2z if T Tga0m

a1m

T Tg0 hyx 1 0 0894ln x 1 p

Bm zTz if T Tg

(31a)

Tg Tg0

s p (31b)

Bs B0se B1s T 273| (31c)

Bm B0me B1m T 273| (31d)

Theparametersfor theCrossmodelandtheTait equation,aswell asthethermalproper-ties,aregivenin Table3.

The thermalconductivity andthermalcapacityareassumedconstantfor eachphase.For the nitrogengasthat is injected,we assumethe relevant propertiesto be constantwithin thewindow of processingconditions;thesepropertiesaregivenin Table4.

16

Table5: Experimentalconditionsfor the gasinjection experimentsin an axisymmetriccylinder.

case temperature pressure initial tdelay

[ C] [105Pa] filling [s]isothermal,Newtonian 171 1.0 67% –isothermal,shear-thinning 179 5.8 67% –non-isothermal 170 5.8 81% 75

4 Experimental validation

For a comparisonof experimentalandnumericalresults,we will focuson the residualwall thicknessasthemostimportantquantity. Furthermore,thestackingof alternatinglyblackandyellow polymerin thecylinder mold (seeFigure3) enablesusto visualizethepolymer flow patternsthroughexperimentalparticle tracking, which can be compareddirectly to thenumericalparticletrackingresults.

4.1 Axisymmetric cylinder

With theaxisymmetriccylinder set-up,threedifferentexperimentshave beencarriedoutundertheconditionsthataregivenin Table5. Whengasis injectedat anoverpressureof1.0bar into thecylinder filled with polystyrenethathasa uniform temperatureof 171 C,the shearrate is low enoughfor the polymerviscosity to be on the Newtonianplateau(seeSection3.3). Hence,if shrinkagewere to be neglected,a residualwall thicknessof approximately36%would beexpected.Theshrinkageof thepolymerwill causethisfractionto besomewhatsmaller. Isothermalshear-thinningconditionsareobtainedwhenthegasis injectedat 5.8baroverpressureinto polystyreneof 179 C. Accordingto [11],shear-thinningwill causetheresidualwall thicknessto decrease(seeFigure2). In thenon-isothermalexperiment,thepolymeris cooledfor 75secondsfrom aninitial homogeneoustemperatureof 170 C beforethe gasis injected. At that time, the wall temperaturehasdecreasedto approximately150 C, which causesa ten-fold increasein the (zeroshear-rate)viscositynearthemold wall comparedto theviscosityin thecenter. Consequently,the residualwall thicknessis expectedto be larger than in the isothermalcases. Thetemperaturesgivenin Table5 areaveragevaluesover thecylinder length,sinceboththetemperaturesatthetopandbottomof thecylinderwereabout5 C lowerthanatthecentralpart. Simulationsshowedthat thesetemperaturedifferencesdid not have any significanteffecton theresults.

Fiveexperimentswerecarriedout for eachcaseof experimentalconditions.Thespec-imenswerecut in half, afterwhich theresidualwall thicknesswasmeasuredwith a mi-crometerat 10mm intervalsalongthecylinder lengthat bothsidesof thegascore.

The simulationresultsfor the residualwall thicknessfraction ε andthe filling timearecomparedwith the experimentalresultsin Figures5 and6, and in Table6. Apartfrom someminor entranceeffects,theagreementfor theresidualwall thicknessesin theisothermalcasesis good. Also the filling time is predictedwell. The slight deviationof the resultsfor the Newtoniancasefrom the valueof ε 0 36 canbe accountedfor

17

Table6: Experimentalandnumericalresultsfor gasinjection in thecylinder mold. Theresidualwall thicknessis averagedover thedomain30mm z 70mm, wherez is theaxialdistancefrom thegasinjectionpoint.

case residualwall thickness[%] filling time [s]experiment simulation experiment simulation

isothermal,Newtonian 34.6~ 0.8 33.8 92 ~ 15 98isothermal,shear-thinning 31.8~ 0.9 32.0 3.2~ 0.5 4.7non-isothermal 54.0~ 1.2 45.8 13.1~ 5.1 29

by theshrinkageof thepolystyrene.Shear-thinningdoesindeedcausea slight decreasein the residualwall thickness,which is in accordancewith theconclusionsof [11]. Thefilling time for thiscaseis muchshorterdueto thehigherpressure,andsomediscrepancybetweentheexperimentalandthecomputedvalueis found.

Onecancalculatethat,for thisgeometry, thesecondarygaspenetrationdueto shrink-agein the packingstagehasa marginal effect on the residualwall thickness(seeAp-pendixB).

The differencebetweenexperimentandsimulationis larger for the non-isothermalcase,althoughthe generaltrendis predictedwell (seethe solid line in Figure6(c)). Ifthis discrepancy betweenmeasuredandcomputedresidualwall thicknessweredueto alag betweenthemeasuredandtheactualmold wall temperatures,thenFigure6(c) wouldsuggeststhat the actualwall temperatureshouldbe lower. Simulationshows thata 5 Cdecreasein wall temperaturedoesindeedyield alargerresidualwall thickness,althoughitcoversonly half theoriginalgapbetweennumericalandexperimentalvalues.Prescribinganactualwall temperaturein thesimulationsthat is about10 C lower thanthemeasuredtemperaturemight yield coincidingexperimentalandnumericalresults,but sucha largetemperaturedifferenceis very unlikely to occurin theactualexperiment.Moreover, the5 C decreasein wall temperaturedoublesthecalculatedfilling time, which wasalreadytwiceaslongastheexperimentalfilling time.

Closerexaminationof the specimensfrom the non-isothermalexperimentsrevealedthattheexperimentalgascontourssometimeswerehighly irregular, with polymerindenta-tionsprotrudinginto thegascore;thiscausestheexperimentalerrorin thenon-isothermalcaseto be larger than in the isothermalcases(comparethe error barsin Figure6(c) tothosein Figures6(a)and6(b)). To explain this, we recall that thecylinder hadbeenini-tially filled with a stackof polystyrenepills. Stackingthesepills is believedto give riseto contaminationsandsmall air gapsat the interfacesbetweenthesepills. It seemsthatunderhigh sheardeformationsuchasshown in Figure5(c), the material(partly) losesits coherenceat theinterfacebetweentwo pills. Thegasthensupposedlybreaksthroughthe polymerpill that coversthe gasfront. Consequently, the numberof polymerlayersdownstreamof thegasfront decreases,which canbeseenin the left half of Figure5(c):of theoriginal 16 coloredlayers,only tenarefoundbetweenthegascoreandthe top ofthespecimen.Theremainingsix layersarevisible asalternatinglight anddarkshadesofgrey at the surfaceof the gascore. This ‘local breakthrough’effect is accompaniedbylocal increasesin wall thicknessat the transitionsbetweenpills, which is shown for anextremecasein Figure5(d). In this respect,we suggestthat the experimentalvaluesin

18

(a) (b) (c) (d)

Figure5: Comparisonof experimental(left) andnumerical(right) flow patternsfor gasinjection into an axisymmetriccylinder; (a): isothermalNewtoniancase,(b):isothermalshear-thinning case,(c): non-isothermalcase,(d): a specimenex-hibiting severe‘local breakthrough’of polymerpills (non-isothermalcase).

19

z

ε

100806040200

0.8

0.7

0.6

0.5

0.4

0.3

0.2

(a) Isothermal,Newtonian.

z

ε

100806040200

0.8

0.7

0.6

0.5

0.4

0.3

0.2

(b) Isothermal,shear-thinning.

z

ε

100806040200

0.8

0.7

0.6

0.5

0.4

0.3

0.2

(c) Non-isothermal.

Figure6: Comparisonof experimentalandnumericalresultsfor gasinjectioninto anax-isymmetriccylinder: relative residualwall thicknessε alongtheaxial distancez from the gasinjection point (error bars= experimentalresults;solid line =numericalresult;dashedline: Newtonian36%limit). Thedash-dotline in (c)hasbeenobtainedby assuminga5 C lowerwall temperature.

Figure6(c) beregardedasan‘upperbound’ for theresidualwall thickness.Apparently,the breakthroughof polymer layersdecreasesthe flow resistance,andhencedecreasesthe experimentalfilling time, which is lessthan half the computedfilling time. Localbreakthroughof polymerlayershasalsobeenfoundin theisothermalexperiments,but toamuchlesserextent,andusuallyfor z 90mm, sothattheinfluenceon theresidualwallthicknessis verysmall.

4.2 Plaque-with-rib

In theplaque-with-ribexperiment,nitrogengaswasinjectedatanoverpressureof 3.0barinto polystyrenehaving a homogeneoustemperatureof 180 C. This gaspressurewasmaintainedduring the cooling of the mold. Eventually, four specimenswereobtained

20

2KK W

rATA;? 7[TA?G ?TTX Q¡Q¢ £¤

Figure7: Locationsof residualwall thicknessmeasurementsfor theplaque-with-ribspec-imens: top view (top) andcross-sectionalview (bottom). Note that for everycrosssection,two rib bottomthicknesses(positions3 and6) andfour rib flankthicknesses(positions1, 2, 4, and5) aremeasured.

from the isothermalexperiments. Thesespecimenswere cut into ten slicesalong thelengthdirection,afterwhich theresidualwall thicknessesat thecenterof therib bottomandthe rib flanks(indicatedin Figure7) weremeasuredwith a micrometer. Theset-updid not allow thefilling timesto bemeasuredaccurately;roughly, they wereof theorderof 30seconds.

Themoldgeometryandcomputationalmeshfor thesimulationof thisexperimentaredepictedin Figure4(b). This meshis rathercoarseascomparedto the meshesusedinthe previous sections.Yet, it consistsof 896 brick elements,which yield nearly30000degreesof freedomfor theStokesequations.It tookapproximately21

2 daysof CPU-timeto simulatetheexperimenton aSiliconGraphicsworkstationwith anR10000processor.

The numericallyandexperimentallyobtainedgascorecontoursfor this experiment,asshown in Figures8 and9, appearto matchwell. Thenumericalresultsalsodemonstratethatthecontributionof thepolymershrinkage(alsocalled‘secondarygaspenetration’)tothefinal gascoresizeprovesto be significant: this is indicatedby theareabetweenthesolid linesandthegrey (polymer)areain Figure8. In Figure10 andTable7, theresidual

21

Table7: Experimentaland numericalresidualwall thicknessfor gasinjection into theplaque-with-ribmold. The valuesareaveragedover the domain12mm y 84mm, wherey is theaxialdistancefrom thegasinjectionpoint.

position residualwall thickness[%]experiment simulation

rib bottom 26.7~ 0.7 21.0rib flank 24.3~ 0.6 23.7

wall thicknesseshave beenrelatedto the hydraulicradiusof the trianglethat makesuptherib. Thewigglesin thesimulationresultsaredueto thecoarsenessof themesh.Thereis a goodquantitativeagreementbetweennumericalandexperimentalresults.Theslightunder-predictionof therib bottomwall thicknessmaybeattributedto themesh,which iscoarserneartherib bottomthanneartherib flank (seeFigure4(b)).

4.3 Discussion

Both isothermalandnon-isothermalgasinjection experimentsin a cylinder mold werecarriedout to validateour modelfor gas-assistedinjectionmolding. Thesimulationsofthe isothermalcasesyielded resultsthat agreedwell with the experimentalresults. Inparticular, the effect of shear-thinning viscositybehavior on the residualwall thicknesswaspredictedcorrectly. For thenon-isothermalexperimenta qualitative agreementwasfound: mold cooling wasseento increasethe residualwall thickness,asexpected.Thequantitativediscrepancy betweenthenumericalandexperimentalresultsfor thiscasecan,to a largeextent,beattributedto experimentalproblems.

The simulationof an isothermalgasinjection experimentin a plaque-with-ribmolddemonstratesthe model’s ability to dealwith three-dimensionalgeometries.Moreover,polymershrinkagewasfoundto have asignificanteffect on thefinal sizeof thegascore.Unfortunately, experimentaldifficulties relatedto the unstablenatureof gaspenetrationprohibitedreproduciblenon-isothermalexperimentsto beexecutedwith theplaque-with-rib set-up.To facilitatesuchexperiments,aninjectionmoldingmachinewith agasinjec-tion unit is required. However, our main conclusionsregardingthe modelingof GAIMarenot affectedby theabsenceof a fully three-dimensional,non-isothermalgasinjectionexperiment,sincethemodeldoesnot treattwo-dimensional(axisymmetric)gas-assistedinjectionmoldingfundamentallydifferentfrom its three-dimensionalcounterpart.

This leadsusto thethree-dimensionalnatureof thegaspenetrationphenomenon.Theresidualwall thicknessfractionis obviously theresultof a forcebalancebetweenthegaspressureandthe(viscous)stressesin thepenetratedliquid. Considerthepenetrationof aninviscidgasinto anaxisymmetriccylinderfilled with aviscousliquid. For Newtonianliq-uids,theresidualwall thicknessfractionis approximately0.36,irrespective of theliquidviscosity(or, equivalently, irrespective of thegaspressure).If thecylinder is filled witha shear-thinning fluid, the residualwall thicknessbecomesa function of the power-lawexponent,asis shown by thesimulationresultsin Figure11 andby theobservationsofPoslinskiet al. [11].

Theresidualwall thicknessε cannotbederivedfrom theshearstressdistribution far

22

(a) (b) (c)

24mm

60mm

96mm

Figure8: Isothermalgas injection simulation for the plaque-with-rib. (a): Simulationresultfor crosssectionx 0mm. (b): Simulationresultfor crosssectionz 2mm. (c): Photographof thebottom(z 0mm) of a transparentspecimen(thenearlyhorizontalline just above thegasbubblecontouris a crack). The linesin (a) and(b) depictthegasbubblecontoursat theendof thefilling stage,i.e.,beforeany shrinkagehasoccurred.

23

(a)y 96mm

(b) y 60mm

(c) y 24mm

Figure9: Isothermalgasinjectionfor theplaque-with-rib:comparisonof simulation(left)andexperimental(right) resultsat differentcrosssectionsperpendicularto they-direction(correspondingto thehorizontallinesin Figure8).

24

y

ε

120100806040200

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(a)Rib bottom.

y

ε

120100806040200

0.7

0.6

0.5

0.4

0.3

0.2

0.1

(b) Rib flank.

Figure10: Comparisonof experimentaland numericalresultsfor gas injection into aplaquewith a triangular: residualwall thicknessε as a fraction of the hy-draulic radiusof the rib triangle(= 4.08mm) alongtheaxial distancey fromthegasinjectionpoint(errorbars= experimentalresults;solidline = numericalresult).

downstreamof the gasfront (which canbe calculatedfrom the Hagen-Poiseuilleequa-tion), sincethat distribution doesnot dependon the constitutive model for the viscos-ity. Consequently, ε is determinedby the flow field at the gasfront, which is three-dimensionalasadvocatedin the introduction.Onemay try to predictthis quantityfromthe two-dimensionalPoiseuilleflow characteristicsof the penetratedliquid downstreamof the gasfront. However, suchattemptsareboundto leadto empirical relations(see,e.g., [11, 24, 25]), which maynot begenerallyapplicable.For instance,Poslinskiet al.proposesa relationthathasthreematerial-dependentparameters[11]. Thosewho persistin searchingfor anempiricalrelationshouldrealizethat thekey parameteris theviscos-ity (or actually: theviscositydistribution),sincethis is theonly parameterleft to vary intheStokesequationthatgovernsthegaspenetrationproblem(equation6). Nevertheless,as long asa reliableempirical relation is not available,accuratepredictionsof the gasdistribution in GAIM productsdo requirethree-dimensionalsimulations.

5 Conclusions

A three-dimensionalmodelhasbeendevelopedfor thesimulationof gas-assistedinjectionmoldingprocesses,andhasbeenimplementedin a finite elementpackage.It is basedona physical,ratherthanon an empiricalapproach.To avoid elaboratethree-dimensionalremeshing,apseudo-concentrationmethod(or: fictitiousfluid method)hasbeenadopted,which employs a material label parameterto distinguishthe polymer from the gas. Itwaspreviously demonstratedthat the model is ableto qualitatively predicta numberofcharacteristicGAIM phenomena[12].

For the isothermalgasinjection experiments,an excellentagreementwasfound be-tweentheexperimentalandthenumericalresults,underbothNewtonianandshear-thinning

25

n

ε

10.90.80.70.60.50.40.30.2

0.37

0.36

0.35

0.34

0.33

0.32

0.31

0.3

Figure11: The effect of the power-law exponentn on the residualwall thicknessε forisothermalgaspenetrationinto anaxisymmetriccylinder (resultsfrom simu-lations).

conditions. For the non-isothermalcase,therewasa qualitative agreementin the sensethattrendin theresidualwall thicknessprofile waspredictedcorrectly, but thecomputedresidualwall thicknesswas smaller than the actualexperimentalvalue. However, theexperimentallyobtainedresidualwall thicknesssufferedfrom a particularexperimentalerror, for which aplausiblecausehasbeengiven.

Due to experimentaldifficulties, which are relatedto the sensitivity to instabilitiesof thegaspenetration,only isothermalgasinjectionexperimentswerecarriedout at theplaque-with-ribmold. Onceagain,thegoodagreementbetweenexperimentsandsimu-lationsfor theplaque-with-ribmold showed,that themodelis indeedcapableof dealingwith a typical, three-dimensionalGAIM geometry.

Thevalidationexperimentsdescribedin Sections3and4weredesignedto separatetheinfluencesof theparametersthatgovern the residualwall thickness(mostnotablyshearrateandtemperature).For thispurposethoseexperimentsservedwell. It is recommendedto furtherevaluatethecomputationalmodelby extendingthesimulationsto GAIM underthe ‘practical’ conditionsof combinedhigh temperaturegradients,high shearrates,andhighpressures.In thatway, theinfluenceof theprocessparameterscanbestudied.

In conclusion,themodelthathasbeendevelopedfor gas-assistedinjectionmolding,canpredict the final gasdistribution in a productandenhancethe understandingof theprocess.In contrastwith othermodelsthathavebeenreportedin theliterature,thismodelyieldsthegaspenetrationfrom theactualprocessphysics(notfromapresupposedgasdis-tribution),is ableto dealwith thethree-dimensionalcharacterof theprocess,incorporatestemperatureeffectsandgeneralizedNewtonianviscositybehavior, andhasbeenvalidatedexperimentally. As such,this modelmeetstherequirementsfor successfulsimulationsofindustrialgas-assistedinjectionmoldingprocesses.

Oncethegasdistribution in a GAIM productcanbesimulatedsuccessfully, thenextimportantstepwill be thepredictionof productproperties[14]. Sinceit hasbeenadvo-catedthatthereductionof sink marksandof residualstressesarethemainadvantagesofGAIM over conventionalinjectionmolding, it will beobviousthat thesearetheproduct

26

propertiesto befocusedon. This topic is presentlystudiedin our laboratory.

Acknowledgement

This researchwas financially supportedby the GraduateSchoolPolymerTechnologyNetherlands.

A Notation

General notation

a α A scalars(regularLatin andGreeksymbols)a˜

columnA matrix (uprightregularLatin capitals)a vector(boldLatin symbols)A α secondordertensors(boldLatin capitals,boldGreeksymbols)

Latin symbols

c – materiallabelcp Jkg 1K 1 specificheatcapacitye Jkg 1 specificinternalenergye m3 computationalelementvolumehr Jkg 1 reactionheatH m characteristicthicknessL m characteristiclengthn – power-law exponentp Pa pressurer Wkg 1 thermalradiationRc s 1 reactionratet s timeT K temperatureTg K glasstransitiontemperature∆T 0 K characteristictemperaturedifference

U ms 1 characteristicvelocityv m3kg 1 specificvolume

Greek symbols

α K 1 linearthermalexpansioncoefficientε – characteristicheight-to-lengthratioε – residualwall thicknessratioγ Nm 1 interfacialtensionΓ boundaryη Pas shearviscosity

27

λ Wm 1K 1 heatconductioncoefficientφ – cross-sectionalliquid fraction;shapefunctionρ kgm 3 densityτ s characteristictimeψ – (pressure)shapefunctionΩ computationaldomain

Vectors and tensors

0 – null vectoror tensorD s 1 rate-of-deformationtensorg ms 2 gravitationalaccelerationh Wm 2 heatflux vectorn m normalvectoru ms 1 velocity vectorx m positionvectorσ Pa Cauchystresstensorτ Pa extra stresstensor

Dimensionless numbers

Br Brinkmannumber= viscousdissipationheatconduction

Ca Capillarynumber= viscousforceinterfacialtensionforce

Fo Fouriernumber= elapsedtimecharacteristiccoolingtime

Gc Gay-Lussacnumber= (characteristicthermalexpansion) 1

Pe Pecletnumber= heatconvectionheatconduction

Re Reynoldsnumber= inertiaforceviscousforce

Sr Strouhalnumber= instationaryinertiaforcestationaryinertiaforce

Superscripts e| with respectto asingleelementn at time level n

dimensionlessvariable

Subscripts

e entrance

i (or j ) ith (or j th) componentof acolumnor matrixn in normaldirectiont in tangentialdirection(s)v air vent;viscousw wall0 characteristicvalue;referencevalue

28

Operators and functions

a materialtimederivativeof a∂a∂t spatialtimederivativeof a∆a differencein a∇ m 1 gradientoperator3 a 3 absolutevalueof aAd deviatoricpartof tensorA (Ad A 1

3trA )

trA trace(or first invariant)of tensorA

II A secondinvariantof tensorA

B Secondary gas penetration in a cylinder

For the cylinder geometry, we canestimatethe differencein residualwall thicknessbe-tweentheprimaryandthesecondarygaspenetrationfrom thefollowing calculation:

Assumea shrinkageof 5% anda residualwall thicknessfraction after primary gaspenetrationof 0.35. For the cylindrical testsample,R is the (fixed) outer radius,r1 isthegaschannelradiusafterprimarygaspenetration,andr2 is thegaschannelradiusaftersecondarygaspenetration.For acircularcross-sectionof acylindrical sampleof lengthL,thevolumesV1 andV2 of thepolymermaterialafterprimaryandsecondarygaspenetrationarerespectively givenby:

V1 π ¥ R2 r21 ¦ L

V2 π ¥ R2 r22 ¦ L

Dueto 5% shrinkage,V2 0 95V1, hence:

V2 π ¥ R2 r22 ¦ L 0 95π ¥ R2 r2

1 ¦ L Simplealgebraicmanipulationyields:

r2 ¨§ 0 05R2 0 95r21

Given that R 8 5 mm (seeFigure4), r1 5 525mm andfrom the above calculation,r2 5 72mm.

Hence,thedifferencebetweenprimaryandsecondaryresidualwall thicknessismerely0 2 mm, which is too smallto indicatein Figures5 and6.

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29

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